Modeling Damped Lyman-alpha Absorbers

Astronomy 626: Spring 1997

In a recent paper, Prochaska &amp Wolfe (1997) present a kinematic
model of damped Lyman-alpha absorbers as thick rotating disks of
individual clouds. Their model treats the random velocities of the
clouds and the thickness of the disk as independent parameters. The
goal of this project is to develop a simple dynamical model for
damped Lyman-alpha absorbers, in which cloud velocities consistent
with equilibrium are determined from the moment equations.

where rho_0 is the central cloud density, R_d is the
disk's radial scale length, and h is the disk's vertical
scale height. The disk is assumed to rotate with constant circular
velocity v_rot; in addition, individual clouds are assigned
isotropic random velocities with 1-D dispersion sigma_cc.

To reproduce the complex profiles and velocity widths of metal
lines in damped Lyman-alpha systems, sight-lines through the disk must
intersect clouds with a wide range of velocities. Such sight-lines are
rare if the disk is thin; thus Prochaska &amp Wolfe favor thick disk
models with h = 0.3 R_d. But a thick disk must be supported
against collapse along the z direction. If individual clouds
are assumed to move ballistically then they need vertical velocities
large enough to reach distances of order h above the disk's
mid-plane. Thus the disk's vertical scale height and random velocity
dispersion can't be set independently; Prochaska &amp Wolfe's adopted
dispersion sigma_cc = 10 km/s seems too small to support the
thick disks they use to match the observations.

This project will develop a simple dynamical disk model of for
damped Lyman-alpha absorbers and implement a Monte-Carlo program to
generate metal-line absorption profiles for this model. A consistent
relationship between disk thickness and velocity dispersion is the
primary goal. Velocity moment equations offer a fairly direct way to
approach to this requirement. To keep things simple, a thin-disk
approximation will be used, and stretched to treat thicker disks.

2. Disk Model

The model to be developed is closely based on the one by Prochaska
&amp Wolfe. Some key assumptions, adopted to simplify matters, are
that

the dynamics of the disk are adequately described by the
thin-disk and epicyclic approximations,

the circular velocity in the disk plane is constant and equal
to v_c,

the disk is vertically self-gravitating, and

the shape of the velocity ellipsoid is constant.

The disk's density profile is similar to Eq. (1), but the vertical
density profile is given by the law for an isothermal self-gravitating
slab (BT87, Prob. 4-25); thus

-R/R_d 2 z
(2) rho(R,z) = rho_0 e sech (-----) ;
2 z_0

note that for large z, Eq. (2) falls off exponentially, with
scale height z_0. Random velocities of individual clouds are
drawn from an anisotropic distribution of the form

(3) f(v) = g(v_R, sigma_R) g(v_phi-v_d, sigma_phi) g(v_z, sigma_z) ,

where g(x, sigma) is a gaussian with dispersion
sigma and mean of zero, sigma_R, sigma_phi,
and sigma_z are dispersions in the radial, azimuthal, and
vertical directions, and v_d is the net rotation speed of the
disk, which differs from v_c due to asymmetric drift.

3. Moment Equations

sigma_R, sigma_phi, sigma_z, and
v_d are functions of position, to be determined from velocity
moments of the Collisionless Boltzmann Equation. The steps are:

For the sech^2(z/2z_0) vertical density profile adopted in
Eq. (2), the vertical dispersion will be independent of z;
thus sigma_z = sigma_z(R); since the velocity ellipsoid has
the same shape everywhere, all three dispersions scale together (NB:
step #3 above explicitly proves that sigma_phi/sigma_R is a
constant). The rotation velocity is also a function of R
only.

4. Monte-Carlo Simulation

Given expressions for sigma_R, sigma_phi,
sigma_z, and v_d as functions of R
simulates the line profile due to a population of clouds distributed
according to Eqs. (2) and (3) along a given sight-line through the
disk. Let the sight-line be specified by the radius R_s at
which it crosses the disk mid-plane and a unit vector n
which points in the viewing direction. The steps are:

chose cloud positions along the sight-line by treating
rho(R,z) as a probability density;

for each cloud, chose velocity components by treating
f(v) as a probability density;

project each cloud's velocity v along
n to determine its line-of-sight velocity;

combine gaussian absorption profiles for individual clouds to
obtain a curve showing total optical depth as a function of
line-of-sight velocity.

5. Strategy &amp Tactics

The moment equation solutions and the Monte-Carlo implementation
are largely independent of one another; thus two teams of three each
can work on these aspects of the project simultaneously, combining
their work at the end. If possible, each team should include someone
taking the QSO absorption-line seminar. The moment equation team will
need good analytical skills, while the Monte-Carlo team will require
some programming expertise. However, the makeup of the teams is left
to the class.

While the moment equation team is working on their part of the
problem, the Monte-Carlo team can design and test the simulation
program, using dummy functions to compute sigma_R(R),
sigma_phi(R), sigma_z(R), and v_d(R). When
the actual expressions for these functions are available, they can be
inserted in the simulation program.

Each team will turn in a written description of their work,
organized as a section to be included in a paper. This description
need not spell out every detail but should enable a reader to
reconstruct what has been done. Also, the moment equation team should
turn in the actual calculations leading up to their result, while the
Monte-Carlo team should turn in a commented listing of the program.
Finally, sample results for various choices of the input parameters
should be provided to illustrate the operation of the program.